Mathematics for the Nonmathematician (Books on Mathematics)

by Morris Kline

BELL, A. E.: Christian Huygens and the Development of Science in the Seventeenth Century, Edward Arnold and Co., London, 1947.

COHEN, I. BERNARD: The Birth of a New Physics, Chap. 5, Doubleday and Co., Anchor Books, New York, 1960.

DAMPIER-WHETHAM, Wm. C. D.: A History of Science and Its Relations with Philosophy and Religion, Chap. 3, Cambridge University Press, London, 1929.

I now propose to set forth those properties which belong to a body whose motion is compounded of two other motions, namely, one uniform and one naturally accelerated; these properties, well worth knowing, I propose to demonstrate in a rigorous manner. GALILEO

To some extent, then, we have come to recognize the broader significance and usefulness of functions and mathematical processes for science in general.

It was again Galileo who perceived the basic principle underlying the phenomenon of curvilinear motion.

In fact, they have become even more common and more complicated in our times, since such phenomena as the motion of bombs dropped from moving airplanes, the trajectories of death-dealing projectiles capable of traveling thousands of miles, and similar problems of modern “civilization,” also fall within the puissance of Galileo’s method.

However, the value of this phase of Galileo’s work is not limited to meting out death and destruction.

Aside from using his results as an illustration of the power of mathematics, we shall see in the space of one chapter how an extension of Galileo’s ideas on projectile motion led, in the hands of Newton, to the greate...

The word formula emphasizes the idea of change because formulas are relationships among variables, and we often like to think of what happens to one variable as another, related variable changes.

Here Galileo made a discovery applying to projectile motion, namely, that one could study its horizontal and vertical motions as though they were occurring separately, and that the position of the bomb at any time could be determined by finding how far it had traveled horizontally and vertically. This idea was new and radical in Galileo’s time.

Aristotle had argued that one motion would interfere with the other, and that only one could operate at any given time. Thus he would have said that the violent motion imparted to the bomb by the airplane would prevail until the acting force was used up, and then the natural motion downward would take over and cause the bomb to fall straight down.

To appreciate how much mathematics accomplishes in this area, one might consider how he would proceed experimentally to find, for example, the dependence of the range of a projectile on the angle of fire. One would have to fire at least dozens of projectiles at different angles, making certain that other factors, such as the velocity with which the projectiles are fired, the shape of the projectiles, and the state of the atmosphere, are constant, and accurately measure the range and angle each time. With all these precautions taken and the information secured, the experimenter might obtain some limited results.

While Galileo was fashioning the new science of motion, Johannes Kepler was making dramatic contributions to one of the most far-reaching developments in the history of Western civilization. This development was begun by Nicolaus Copernicus and its essence was a radically new mathematical theory of planetary motions.

Copernicus was born in Poland in 1473 and, after studying mathematics and science at the University of Cracow, decided to go to Italy, the center of the revived Greek learning.

Cool. I am reading this in Poland :-)

In Copernicus’ time the theory required a total of 77 circles to describe the motion of the sun, moon, and the five planets known then.

Damn

He also had the humility, patience, and energy to perform extraordinary labor which mark great men.

In his book On the Motion of the Planet Mars, published in 1609, Kepler announced the first two of his three famous laws of planetary motion. The first of these is especially remarkable, for Kepler broke with the tradition held for 2000 years that circles or spheres must be used to describe heavenly motions.

Cannot express the joy of reading such paragraph .

Kepler’s law of equal areas.

Beautiful

Thus he believed that there was a music of the spheres which produced a harmonious tonal effect, not one given off in actual sounds but discernible by some translation of the facts about planetary motions into musical notes.

The work of Copernicus and Kepler is by far the most dramatic, startling, and influential development in the formation of modern culture. The first surprising feature, and one which in itself makes their work astounding, is the sharp break from existing thought.

\o/

But to the people of Copernicus’ time the argument that we do not feel ourselves moving at the very high speeds called for by the new astronomy was convincing.

Consistent fools?

In view of the numerous and sound arguments against the heliocentric theory and the challenge it posed to the prevailing religious thinking of the times, what made Copernicus and Kepler take up this long-discarded thought and pursue it so courageously? For what most other men would call a mess of pottage, they broke with established physics, philosophy, religion, and common sense.

COURAGE

Hence Copernicus and Kepler believed, when each found a more harmonious and simpler theory, that their work was indeed a description of the divine order of things.

We find therefore, under this orderly arrangement, a wonderful symmetry in the universe, and a definite relation of harmony in the motion and magnitude of the orbs, of a kind that it is not possible to obtain in any other way.

What distinguishes their religious convictions from those of their contemporaries is that they did not tie themselves to literal interpretations of the Holy Writings. They searched for the word of God in the heavens.

The core of the argument which Copernicus and Kepler presented for the heliocentric theory was its mathematical simplicity.

Galileo, himself, though he lectured on Ptolemaic theory until 1605, had been converted to Copernicanism by a work of Kepler. In 1611 he openly declared for Copernicanism. His own observations convinced him that the Copernican system was correct, and in the classic Dialogue on the Great World Systems he defended it strongly.

Galileo's rebelion is all over this book, damn good book.

Did Copernicus break completely with Greek astronomy?

In view of the fact that Galileo had discovered the laws which underlie terrestrial motions and Kepler had discovered the basic laws of planetary motion one would expect that the scientists of the seventeenth century would have regarded the theory of motion to be complete. But to scientists who seek the ultimate design of our universe, the two accomplishments we have just described immediately suggested more profound problems.

to derive two or three general principles of motion from phenomena, and afterwards to tell us how the properties and actions of all corporeal things follow from these manifest principles. . .

Then one man smart enough to distinguish the worthy ideas of his predecessors from the welter of suggestions and results and imaginative and audacious enough to fit the significant ideas into a master plan makes the culminating and definitive step.

The elementary education Newton received in local schools of a small English town could hardly have given him much of a start, and in his youth Newton showed no promise.

Here, at last, Newton got the opportunity to study the works of Copernicus, Kepler, and Galileo, and here he had at least one good teacher, the distinguished mathematician Isaac Barrow. His university work was not outstanding and he had, in fact, such difficulties with geometry that he almost changed his course of study from science to law. However, Barrow did recognize that Newton had ability.

Come on... Every author creates its own Newton... In contrast to his science.

Newton apparently was not a successful teacher, for few students attended his lectures; nor did anyone comment on the originality of the material he presented.

The average person relates this fact to the great weight of the automobile. However, weight plays no role here because the force of gravity acts downward and has no effect on motion along the ground. The forceful push is required because the mass resists change in speed. Hence, it is the mass of the automobile rather than the weight which calls for the exertion of great force.

This Kline guy was no average teacher.

Newton conjectured (and later proved) that for purposes of gravitational attraction the mass of the earth could be regarded as though it were concentrated at the earth’s center.

All of the above considerations about weight are now no longer purely academic flights of fancy but are important factors in the process of determining the paths of rockets which are sent out to strike the moon.

The essence of the scientific method created by Galileo and Newton is to establish basic quantitative physical principles and to apply mathematical reasoning to these principles.

That's your homework, Tiger!

We can turn our argument around. We observe that g increases from the equator to the poles. This increase can be explained by assuming that the earth rotates. Hence we have reason to believe that the earth rotates.

We should now recall that one of the major problems challenging seventeenth-century scientists was the question whether the same physical principles could account for terrestrial and celestial motions.

As to heavenly motions, the three famous laws of Kepler, which he had inferred from observations, were seemingly independent of the law of gravitation. The truly great triumph of Newton was his demonstration that all three Keplerian laws were mathematical consequences of the law of gravitation and the two laws of motion.

Today we have almost daily evidence that Newton had found sound physical principles which govern the operation of the universe.

Instead rockets project the satellite upward to a high altitude where the air resistance is negligible; there a mechanism turns the satellite to a horizontal direction and another rocket gives it a horizontal velocity. Then the satellite follows an elliptical path.

Newton went further in his speculations and conjectured that the planets must have been shot from the sun at some angle and, upon reaching their present distances, must have retained enough “horizontal” velocity to start moving in their elliptical paths around the sun. This conjecture is still the accepted theory of the origin of our solar system.

This is cool

Here was a majestic scheme which embraced the fall of a stone, the tides of the oceans, the moon, the planets, the comets which seemed to sweep defiantly through the orderly system of planets, and the most distant stars. This view of the universe came to a world seeking to secure a new approach to truth and a body of sound truths which were to replace the already discredited doctrines of medieval culture.

This is thrilling

Thus it was bound to give rise to revolutionary systems of thought in almost all intellectual spheres. And it did.

Indeed, mathematics offered not merely the vehicle for scientific expression but the most powerful tool for the real work of science, that is the acquisition of knowledge about the physical world and the organization of that knowledge in coherent systems.

Galileo and Newton had set about finding quantitative laws that related matter, space, time, forces, and other physical properties, but had wisely decided not to look into causal relationships; that is, they had deliberately avoided such questions as why bodies fall to earth or why planets move around the sun.

The astronomical work of Copernicus. The books by Armitage, Dreyer, Koyre, Kuhn, Wolf, and any number of others listed in the Recommended Reading would be fine source material.